The Potters have snapped up the custodian from Premier League side Southampton and he joins the club on a loan deal.
Stoke are keen to have an option to keep Gunn on a permanent basis if they are happy with how he performs during the duration of his loan.
And, according to the Sun, they have secured an option to buy in the loan agreement.
The 24-year-old goalkeeper fell down the pecking order at St Mary’s under Ralph Hasenhuttl and has moved on in search of regular game time.
Gunn made the move to Southampton from Manchester City in the summer of 2018, costing the south coast outfit a fee of £13.5m to sign.
He will be looking to hit the ground running with Stoke, who are bidding to fight for promotion back to the Premier League.
The Potters have collected five points from their opening four Championship games and next play Luton Town in an away fixture on Saturday.